Mean value one of prime-pair constants

2008/06/10 by Fokko van de Bult, van de Bult, Fokko, Jaap Korevaar +1
Mathematics · #11P32 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0806.1667

openalex publication_date 2008/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For k greater than 1 and r different from 0, let pi^k_2r(x) denote the number of prime pairs (p,p^k+2r) with p not exceeding (large) x. By the Bateman-Horn conjecture, the function pi^k_2r(x) should be asymptotic to (2/k)C^k_2rli₂(x), with certain specific constants C^k_2r. Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy-Littlewood constants C_2r for prime pairs (p,p+2r). The conjecture is supported by extensive numerical work.

Citations

• Prime pairs and Zeta's zeros

Related

• Averaged Form of the Hardy-Littlewood Conjecture
• Lower bound for the remainder in the prime-pair conjecture
• On primes of the form n₁^u + n₂^v + k, on average
• The error term in counting prime pairs
• Prime pairs and Zeta's zeros